JEE Main 2022 — Hyperbola Question with Solution

An ellipse $$E:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$ passes through the vertices of the hyperbola $$H:\frac{x^2}{49}-\frac{y^2}{64}=-1$$. Let the major and minor axes of the ellipse $$E$$ coincide with the transverse and conjugate axes of the hyperbola $$H$$, respectively. Let the product of the eccentricities of $$E$$ and $$H$$ be $$\frac{1}{2}$$. If $$l$$ is the length of the latus rectum of the ellipse $$E$$, then the value of $$113l$$ is equal to _____________.